23 December 2007

Reciprocal fuel economy

You save more fuel switching from a 15 to 18 mpg car than switching from a 50 to 100 mpg car.

This sounds counterintuitive at first. But the "natural" units for fuel consumption, at least in this case, are not distance per unit of fuel but units of fuel per distance. In some parts of the word fuel usage is given in liters per 100 km; let's say we were to give fuel usage in gallons per 100 miles. (The constant "100" is just there to make the numbers reasonably sized.) Then switching from a 15 mpg car to an 18 mpg car is switching from a car that gets 6.67 gal/100 mi to 5.56 (lower is better); switching from a 50 mpg car to a 100 mpg car is switching from a car that gets 2.00 gal/100 mi to 1.00. (Another interesting consequence -- switching from 50 mpg to 100 mpg has the same effect as switching from 100 mpg to ∞ mpg, i. e. a car that uses no fuel at all.)

The idea is that chopping off the low-fuel-economy tail of the distribution (by legal means) would be a much easier way to reduce oil consumption than trying to make very-high-fuel-economy cars.

But not all incremental achievements are created equal. It was probably a lot harder to get from 1 mpg to 2 mpg than it will be to get from 100 mpg to 101 mpg.

Also, note that a pair of cars that get, say, 20 mpg and 50 mpg will average "35 mpg" in the way that the new regulations for average mileage of a automaker's fleet are written; but for each car to go 100 miles, it'll take a total of seven gallons of fuel, for a fuel economy of 200/7 = 28.6 mpg. (This is the harmonic mean of 20 and 50.) The regulations aren't necessarily flawed -- they probably should be stated in terms of the measures of fuel economy that are most commonly used -- but there's room for possible misunderstanding.

Another place I can think of where the "natural" units are the reciprocal of the ones that are habitually used is in statistical mechanics; there are tons of formulas there that have temperature in the denominator, and for the purposes of statistical mechanics it makes more sense to use inverse temperature. (I've written about this before, I think; it basically comes out of the fact that the partition function involves inverse temperature.) Are there others?

The weird thing in the potato paradox isn't how much weight is lost. The "paradox" relies on the fact that you're intuitively thinking of percent-by-volume while the problem really means percent-by-weight.